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Bark estimation : bark volume and bark taper Grewal, Harjit Singh 1980

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BARK ESTIMATION: EARK VOLUME AND BARK TAPER by HARJIT SINGH GREWAL B.Sc.F. , University of Toronto, 1975.. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF \ THE REQUIREMENT FOR THE DEGREE OF MASTER OF FORESTRY in the FACULTY OF GRADUATE STUDIES Department of Forestry We accept this thesis as conforming the required standard TEE UNIVERSITY OF BRITISH COLUMBIA October,1980 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an a d v a n c e d d e g r e e at t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a , I a g r e e t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r a g r e e t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p u r p o s e s may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h C o l u m b i a 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date OcX&for 16. /9fr<Q i i ABSTRACT SUPERVISOR: A. KOZAK The interest i n alternate energy sources has prompted the study of tree bark. This thesis deals with the estimation of tree bark volume and bark taper above ground i n the main stem. A bark volume equation was selected based on low average bias and standard error in a l l the 32 B r i t i s h Columbia tree species groups studied. The bark eguation predicts bark volume with l i t t l e bias in most species with the exception of the very thick-barked ones such as coastal Douglas-fir. The bark volume equation was compared to two other approaches of estimation of bark volume and proved to be s l i g h t l y superior. From the form of bark taper in the tree species studied a dual-eguation system was chosen to predict bark thickness at any pcint i n the main bole of the tree. The system consists of one equation for the top portion and another for the bottom section. The two are joined together and are continous at the i n f l e c t i o n p c i n t i Ihe taper prediction system works well for a l l the 32 species groups considered. However bias i s r e l a t i v e l y higher near the butt of the tree especially in thick-barked species. On the whole the system has the attributes of low bias and standard error. It also proved superior when compared to another method of predicting bark thickness. i i i ACKNOWLEDGEMENT The author wishes to thank his supervisor Dr. .A. .Kozak for suggesting the topic of research and for his guidance and patience throughout the period of wcrk on t h i s thesis.. Also the author thanks Drs. J.P. Demaerschalk and CD. Munrc for reviewing the thesis and providing welcome advice and c r i t i c i s m . Dr. D. Williams and Miss Tina Duke of the Computing Centre helped with the IMT program for producing t h i s document on the computer, and th e i r assistance i s gr a t e f u l l y acknowledged. . Fi n a l l y the author wishes to express h i s gratitude to his friends and acquaintances at O.B.C. who have made graduate studies enjoyable to him. i v TABLE OF CONTENTS TITLE PAGE i ABST B ACT i i ACKNOWLEDGEMENT . i i i TABLE CF CONTENTS i v LIST CF FIGURES ........................................... vi LIST CF TABLES . .. v i i 1. Introduction ........................................... . 1 2. Literature Beview 3 2.1 Bark - The Resource 3 2.2 Bark Volume And Taper ............................... 5 2.3 Measuring Eark Thickness 6 3. Source Of Data 8 4. .Bark Volume Estimation 13 4.1 Objective .......... 13 4.2 Method 13 4.2.1 Variables ... 13 4.2.2 Transformations Of Variables 14 4.2.3 Stepwise Elimination 14 4.2.4 Bias And Standard Error 15 4.3 Results And Discussion 16 4.4 Comparison Of The Eest Eguation Method And The Volume Table Method 22 4.5 Comparison Of Volume Eguation And The Method Of Difference Of P r o f i l e s 26 5. Bark Taper Prediction 28 5.1 Objective 28 5.2 Method 28 5.2.1 I n f l e c t i o n Point 28 5.2.2 Upper Taper Model ............................. 31 5.2.3 Lower Taper Model 32 5.2.4 Characteristics Of The Bark Taper Model 33 5.2.5 Determination Of Model Coefficients C1, C2, C3 And C4 33 5.2.6 Bias And Standard Error ........................ 34 5.3 Results And Discussion 35 5.4 Comparison Of Bark Model With The Method Of Difference Of P r o f i l e s 41 6. Conclusion • 45 LITERATURE CITED 48 APPENDIX I - L i s t Of Symbols ........................... . 50 AEPENDIX II - Comparison Of Total (accumulated) Bias In The 4 Chosen Equations {in Cubic Meters) 51 APPENDIX III - Comparison Of Standard Error Of Estimate In The 4 Chosen Equations (in Cubic Meters) 52 APPENDIX IV - The Regression Coefficients Of Bark Volume Equation {4.4} > 53 AEPENDIX V - Scatterplot Of Relative Bark Thickness Vs. Relative Height For Coastal Mature Douglas-fir (DF1M) .. 54 AEPENDIX VI - Scatterplot Of Relative Bark Thickness Vs. Relative Height For Aspen (A3AA) 55 AEPENDIX VII - The Biases Of The Bark Taper Model For A l l Species Groups (in Centimeters) .. . 56 APPENDIX VIII - Standard Error Of Estimate For The Taper Model (in Centimeters) 57 v i LIST OF FIGURES 1. Figure 1. Schematic Representation Of Bark Thickness Mcdel 29 2. Figure 2..Bark Thickness Prediction Using The Difference Of P r o f i l e s Method 41 v i i LIST OF TABLES 1. Table 1. Description Of Species Groups. . 10 2. Table 2. Range Of Diameter At Breast Height (DBH) And Total Height (HT) . 11 3. Table 3i Range Of Double Bark Thickness At Breast Height (BTBH) And Bark Volume (BVOL) . 12 4. Table 4. Ranks Of Standard Error For The 4 Best Bark Volume Eguaticns In The 32 Species Groups. 17 5. Table 5..Comparison Of Standard Error Of Estimate Of The 4 Eest Bark Volume Equations In Four Selected Species Groups. . 18 6. Table 6. Average Sectional Bias And Standard Error Of Estimate Of Bark Volume Equation For Birch. 20 7..Table 7. Average Sectional Bias And Standard Error Of Estimate Of Bark Volume Equation For Douglas-firi ...... 21 8. Table 8..Comparison Of Bark Volume Equation And The Method Of The Volume Table Estimation. 24 9. Table 9. Comparison Of Bark Volume Equation And The Method Of The Eifference Of The P r o f i l e s . 27 <\0. Table 10. Reqression Coefficients B1, B2 And B3. 36 11. Table 11. Tcp Model Co e f f i c i e n t s C1 And C2 37 12..Table 12. Average Bias And Standard Error Of Estimate Of Bark Taper Model For Birch 39 13. Table 13. Average Bias And Standard Error Of Estimate Of Bark Taper Model For Douglas F i r . 40 14. Table 14. Comparison Of Bark Thickness Using Bark Model And The Method Of Difference Of P r o f i l e s . 42 15. Table 15..Comparison Of Bark Thickness Using Bark Model And The Method Of Difference Of P r o f i l e s 43 1 1. INTEODOCTION The object of t h i s study i s to estimate the volume of tree bark and the manner in which bark thickness varies along the bcle i n ccmmercial B r i t i s h Colombia tree species. . In view of the increasing global demand for energy, tree bark w i l l become an important source of energy i n the near future. Whereas the possible end use of bark l i e s i n the research domain of chemists and engineers, t h i s study i s necessary i n order to estimate the size of the resource available. Eeing faced with the uncertainty of future o i l supplies from the Middle East we should be making use of the resources that are readily available to us. Tree bark i s such a resource.. This study i s biometric i n nature. We would l i k e to relate tark parameters such as bark volume and bark thickness to conventional tree parameters such as diameter at breast height and t c t a l height cf the tree. The objective i s to be able to estimate t o t a l bark volume of a tree and bark thickness at a given height frcm the ground, based on measurements at the breast height level and the height of the tree.. The scope of t h i s study covers a l l coniferous and hardwood ccmmercial tree species of B r i t i s h Columbia, a t o t a l cf 32 species groups. This coverage includes a range of both th i n and thick barked species. This study investigates bark volume equations • for ccmmercial tree species i n B r i t i s h Columbia and develops a taper system to predict bark thickness at any s p e c i f i c position in the tree. Various s t a t i s t i c a l techniques are used including both 2 l i n e a r and non-linear estimation. 3 2. LITERATURE REVIEW 2-1 Eark - The Resource In t h i s age cf the d e c l i n i n g f o s s i l f u e l s u p p l i e s i n t e r e s t i n a l t e r n a t e sources of energy i s mounting. Tree bark, being an abundant and renewable resource with a high heating value (Scheider,1977) , i s a v i a b l e source of energy. In Oregon, f o r example, a commercial process to p e l l e t i z e D o u g l a s - f i r bark has teen developed t o produce a high energy f u e l f o r c o a l and gas a p p l i c a t i o n s ( E l a c k m a n , 1978) . Cf course, bark can be used f o r other purposes b e s i d e s f u e l . Seme s u r p r i s i n g r e s e a r c h r e s u l t s show bark to be p o t e n t i a l l y e f f e c t i v e as : an o i l absorbent , an e m u l s i f i e r , a pulp i n g r e d i e n t , a water p u r i f i e r , an a c i d f i l t e r , a no-resin-added s t r u c t u r a l board (barkboard) , a moulding medium and an a e r a t o r . Eark a l s o has p o t e n t i a l as i n s u l a t i o n board , p a r t i c l e b o a r d , f i r e place b r i g u e t s and as an adhesive (Letkeman, 1977). I t i s a l s o e f f e c t i v e as a highway mulch and i n r u r a l s e p t i c systems (Mater, 1 977) . There are numerous chemical e x t r a c t i v e s found i n the bark of t r e e s i n c l u d i n g many d a i l y use items. Bark i s a l s o used as a grass i n h i b i t o r i n landscape a r c h i t e c t u r e . . The most important s i n g l e use of bark (and wood) i s t h a t of energy. In 1972, n e a r l y h a l f the wood cut f o r man's use was f o r f u e l (PAO, 1974). Worldwide, more people are warmed by wood and bark than by any ether f u e l . In western Canada, r e s e a r c h i n d i c a t e s that D o u g l a s - f i r ( 4 Pseudotsuga menziesii {Mirb.}Franco) bark has a heating value of 5611 Kcal/kg (10,100 Btu/lb), western hemlock ( Tsuga hetercphylla {Baf.}Sarg.) has a heating value of 5,444 Kcal/Kg (9,80C Btu/lb) and western red cedar( Thuja P l i c a t a Donn) bark has a heating value cf 4833 Kcal/kg (8700 Btu/lb) respectively (Dobie and Wright, 1975), It i s i n t e r e s t i n g to note that the heating value of lodgepole pine ( Pinus contorta Dougl.) bark beinc 5997 Kcal/kg (10794 Btu/lb) i s higher than Douglas-fir lark (Chang and Mitchell,1955).. In B r i t i s h Columbia, Smith and Kozak(1967,1971) estimated that 5 m i l l i c n ovendry tons of bark were pote n t i a l l y available from the 1.7 b i l l i o n cubic feet of wood logged at that time. The estimate cf the present a v a i l a b i l i t y of bark i s 7.4 m i l l i o n tons associated with the 2.5 b i l l i o n cubic feet of wood harvested (Smith and S z i k l a i , 1 979) . In circumstances where energy i s more valuable than wood genetic selection could be employed to propagate fas t growing, t h i c k l y larked, in d i v i d u a l s . Genetic selection to pick thick barked indiv i d u a l s f c r propagation could be carried out. 5 2.2 Eark Volume And Taper Interest in estimating bark volume has previously been directed towards density and ovendry weight estimations for i n d u s t r i a l use. Mi l l i k a n (1955) measured bark weight and fuel properties of several eastern Canadian species.. S i m i l a r l y Hale (1955) determined the thickness and density cf bark for 6 pulp«cod species at the Eastern Forest Products Laboratory i n Ottawa* He noted that bark i s not a uniform substance and that i t i s in fact more variable than wood, both i n c e l l structure and chemical composition . The thickness and density of bark are both subject tc variation with the age of the tree.. In general bark i s r e l a t i v e l y thin in young trees and thicker i n older cnes. Few attempts have been made to estimate bark volume.. Meyer (1946) proposed simple formulae for bark thickness and bark volume of the following form: {2. 1} ET = D/2 (1-k) {2.2} BVOL = V (1 - k 2 ) where BT i s bark thickness, BVOL i s bark volume, D i s usually diameter at breast height outside bark DBHOB, but could be diameter outside bark DOB at any point on the tree , V i s the stem volume including bark and k i s the regression c o e f f i c i e n t for the r e l a t i o n cf diameter inside bark (DIB) to DOB. However,as admitted by Meyer i t i s d i f f i c u l t to represent the relationship between bark thickness and diameter by a single 6 straight l i n e ever the entire range of diameters. Therefore i t i s necessary to include other mensurational c h a r a c t e r i s t i c s such as DEH ,height and bark thickness. .Dimitrov (1976) found that the best models for estimating bark volume for spruce species i n Bulgaria were 3 non-linear functions which expressed bark volume as a function of DBH, height, volume, age and s i t e c l a s s . . Johnson (1966) studied young growth Douglas-fir and found that the tark factors for the upper stem are d i f f e r e n t from those for the lower stem and suggested a polynomial for determining the upper stem bark factors. His polynomial i s a function of age ,height, diameter at breast height, bark factor at stump height and diameter at which point the bark factor i s desired. fin unsuccessful attempt was made by Stayton and Hoffman (1970) to develop accurate equations that predict bark thickness at a s p e c i f i c tree heiqht for sugar maple. . 2.3 Measuring Jar k Thickness The common instrument for measurement of bark thickness measurement i s the Swedish bark gauge. This consists of a steel shaft, h a l f - c y l i n d r i c a l i n shape, which i s pushed through the bark. The cutting edge i s d u l l on one side so that the shaft w i l l penetrate the bark but stop at the wood. Doubling the bark thickness and subtracting from the outside dimension gives dianueter inside tark. Problems with dir e c t measurements of bark can be expected f c r mature, thick barked species such as Douglas-fir and care 7 must be taken to ensure accuracy of measurements . . Von Alther; (1964) compared bark thickness measurements taken with Swedish bark gauges and diameter tapes. According to him the common sources cf error were f i r s t l y , the wood beneath i s being deeply penetrated by the gauge, re s u l t i n g i n over-estimation of tark thickness, and secondly, the common practice of pushing the gauge through the bark by means of a whack with the inside of the hand nearly always resulted in a penetration of the wood. Mesavage (1969) traced the sources of error i n bark thickness measurement tc 1) ambiguous gauge readings caused by unevenness cf tark surface at point of measurement , 2) inadeguate sampling,and 3) incorrect seating of the c h i s e l of the bark measuring gauge(usually short of the wood, but occasionally too deep) . 8 3. SOURCE OF DATA The raw data for this study was c o l l e c t e d by the B r i t i s h Columbia Ministry of Forests (Inventory Branch).. A t o t a l of 31,523 trees were measured. The trees belong to 32 species groups according to species,maturity and region. For each tree, the following measurements were recorded: diameter to the nearest 0.1 inch inside and outside bark at 1, 1.5, 2, 4.5 feet above ground and at each tenth of the height above breast height and the t o t a l height. This data was collected by the B.C.F.S. f o r t h e i r tree volume studies. Since the data contained information on both diameter inside and outside bark we were able to use i t to compute bark thickness at the various positions of the tree. Double bark thickness (DBT) i s thus defined as: {.3. 1} LET = DOB - DIE where EIB i s the diameter inside bark and DOB i s the diameter outside bark respectively. The o r i g i n a l data in imperial units were converted to metric units before computations were carried out. The convertion factors used were as follows: 1 inch = 2.54 centimeters 1 f t = 0.3048 meter 1 cu. f t . = 0.02832cu. meter Diameter at breast height(DBH) i s 1.3 meters above ground by convention. This was estimated from diameter at 4.5 feet above grcund by using a non-linear function developed by 9 Demaerschalk and Kozak (1975). The function has the form: k2 {3.2} DBH (1. 3meters) = kl.DBH (4.5feet) where k1 and k2 are c o e f f i c i e n t s derived f o r each species grcup used in th i s study. Eark volume was computed per section of the bole by subtracting the volume with bark from the volume excluding bark; Smalian's formula was used for finding the volume per section. Smalian's formula (Husch et a l . 1970) states that volume, V i s : 1 {3.3} V = (A1 + A 2) 2 where A1,A2 are the cross sectional areas of the two ends of the section and L i s the length of the section. The extent to which bark fissures a f f e c t bark volume was net examined in th i s study. Tables 1 to 3 give the descriptions of the species groups, the range of the height (HT), diameter at breast height (DBH) , bark thickness at breast height (BTBB) and bark volume(BVOL). . 10 Table 1. Description of Species groups. Tree Groups of Interest Number of Species/Maturity/Zones Code Trees Douglas-fir mature zone ABC DFIM 603 Douglas-fir immature zone ABC DF1I 394 Douglas-fir zone DEFGHIJKL DF2A 3000 Eed-cedar mature zone ABC CE1M 595 Red-cedar immature zone ABC CE1I 428 Bed-cedar zone DEFGHJIKL CE2A 1477 Hemlcck mature zone ABC HE1M 1276 Hettlock immature zone A EC HEI I 736 Hemlock zone DEEGHIJKL HE2A 1785 Balsam zone ABC BA 1 A 816 Balsam zone DEFGHIJ BA2A 3000 Balsam zone KL B A3 A 577 Spruce mature zone ABC SP1M 3 53 Spruce immature zone ABC SP1I 318 Spruce zone DEFGHIJ SP2A 3000 Spruce zone KL SP3A 3000 Yellow Cedar a l l zones CYAA 296 White Pine zone ABC PW 1 A 86 White Pine zone DEFGHIJKL PW2A 254 Lodgepole Pine ABCDEFGHIJ PL12 2722 Lodgepole Pine KL PL3A 1364 Yellow Pine a l l zones PYAA 641 Larch zone AECEEFGHIJ L12A 756 Larch zone KL L3AA 203 Cottonwood zone ABCDEFGHIJ CT12 322 Cottonwood zone KL CT3A 550 Alder a l l zones D123 519 Maple a l l zones MEAA 139 Birch zone AECEEFGHIJ BI12 302 Birch zone KL BI3A 193 Aspen sone ABCDEFGHIJ A12A 768 Aspen scne KL A3AA 1050 Total 31523 11 Table 2. Range of Diameter at Breast Height(DBH.) and Tree Height (HT.) Range of DBH(in cm) Range of HT(in m) Cede min avg max min a wg max DFIM 15.8 73.7 216. 8 9.23 38.97 76.71 DF1I 5.3 28.3 84. 6 6. 27 24.81 51.54 DF2A 2.0 35.7 136. 0 2.62 22.42 49.43 CE1M 12.9 6 1.9 216. 6 7.31 30.77 63.45 CE1I 3.3 24.8 93. 6 4.05 17.92 46. 17 CE2A 4. 5 35.2 156. 7 3.26 21.03 51.26 HE 1M 9.6 42.2 190.0 5.57 28.43 63.33 HE1I 5.0 20.9 85. 5 4.99 20. 1 1 48.46 HE2A 3.7 28.8 135. 2 3.20 22.37 54.37 E A 1A 10. 4 45.7 147. 9 4.87 30.53 59.77 E A 2 A 3.7 23.4 89.0 3.20 18.31 48. 37 E A 3 A 12. 9 28.2 64. 4 9.20 20.51 32.67 SE 1M 14.5 88.8 376. 9 12. 31 46.62 76.96 SP1I 3.0 36.6 106. 4 3.29 28. 10 51.99 SP2A 4.0 32.2 111.1 3.01 23. 18 54.46 SP3A 8.0 27. 1 77. 7 7.19 21.56 43.46 CYAA 8. 4 32.0 124. 8 5. 73 18.99 45.23 EW 1A 8. 2 44.7 116. 5 11.64 31.34 58.06 EW2A 7. 1 37.9 105.0 6.76 30.17 53.70 E112 3.7 23.9 59.8 3. 13 21.55 39.65 EI3A 4.3 23.3 50. 6 6.09 19.26 30.81 EY.AA 3.0 38.8 112.8 2.98 20.82 48.64 L12A 4. 1 34.7 89. 3 4.20 26.85 53.67 13 A A 8.2 20. 1 43. 2 8.44 17.55 28.19 CT12 6. 6 31.7 128. 2 9.14 24.68 48.85 CT3A 8.9 36.4 95. 9 13.13 28.22 4 6 i 11 D123 6.2 21.3 54. 4 8.16 20.28 32.76 ME A A 7.0 17.2 41.6 11. 36 18.20 30.02 BI12 5.0 19.9 53. 7 7.98 19.01 27.18 EI3A 12.9 20. 6 39.8 13.59 20.51 24.04 A12A 3.7 19. 3 54. 2 • 4.35 18.66 32. 24 A3 AA 9.3 26.7 72.8 10.30 22.35 34.56 Table 3. Range cf Double Eark Thickness at Breast Heigiit (BTBH) and Bark Volume (BVOL). Range of ETBH (in cm) Code min a vg max DF 1M 1.7 12.4 30.4 DF1I 0. 2 3.5 14.9 DF2A 0. 2 5.4 30.7 CE1M 0.5 2.6 9.6 CE1I 0.2 1. 1 3. 5 CE2A 0. 2 2.2 11.1 HE 1M 0.7 2.9 9.6 HE1I 0.2 1 .4 5. 8 HE2A 0.2 2.5 9. 1 E A 1 A 0.5 2.0 6.0 EA2A 0.2 1.4 10. 1 B A3 A 0.5 1.6 5.0 SF 1M 0.7 2.4 7.3 SP1I 0. 2 1.4 5.0 SP2A 0.2 1.5 11.4 SP3A 0.5 1.3 5.0 CY AA 0. 2 1.1 4.0 PW 1 A 0.5 2.0 5.5 EW2A 0.5 1.9 4. 5 P112 0.2 1.2 5.3 PL 3 A 0.2 0.9 2. 5 PYAA 0. 2 5. 5 1 5. 4 L12A 0.5 5.4 27.9 13 A A 0.7 1.3 2.2 CT12 0.5 3.2 16.2 CT3A 0.5 4.3 12. 1 D123 0.2 1.0 3.0 MBA A 0.5 C.9 2. 5 BI12 0. 2 1. 1 2.7 BI3A 0.7 1. 1 2.2 A12A 0.2 1.6 6.8 A3 AA 0.5 2.0 8.3 Range of BVOL (in m3) min avg max .025 2.550 23.580 . 002 0. 225 2. 230 .000 0.374 6. 350 .005 0.620 5. 720 .000 0.065 0. 860 . 000 0. 245 4. 1 80 .008 0.460 4. 897 .002 0.073 1. 260 . 000 0.230 4. 530 . 004 0. 440 3.390 . 002 0.082 1. 600 .010 0.0 94 0. 530 .012 0. 908 5. 320 .000 0. 151 1. 160 .000 0. 134 2.610 .005 0.081 0. 566 . 003 0. 162 2. 460 . 005 0. 347 2.640 . 002 0.216 1. 300 .000 0.060 0. 318 .002 0. 044 0.297 .000 0.437 3. 300 . 001 0.392 2. 450 .005 0.041 0. 152 .003 0.337 4. 190 .004 0.410 3. 450 . 001 0.055 0. 329 .003 0.029 0. 206 .001 0.039 0. 217 .011 0.042 0. 123 .000 0.077 0. 554 .007 0. 108 0. 734 1 3 4 . BARK VOLUME ESTIMATION 4 . 1 Objective The objective of this study i s to obtain main bole bark vclume regression eguations that w i l l adequately describe a l l the common tree species in B r i t i s h Columbia. The c r i t e r i a for choosinq the independent variables must be low bias and standard error cf estimate. 4.2 Method 4 . 2 . 1 Variables The dependent variable i s hark volume, BVOL (in m 3), and the independent variables are diameter at breast height,DBH (in cm), t o t a l height cf the tree,HT (in m), and bark thickness at breast height, BTBH (in cm)..The dependent variable, bark volume, was plotted against each of the independent variables. From these plots i t appeared that logarithmic transformation would make the variance uniform. Base 1 0 logarithm was a r b i t r a r i l y chosen for these transformations.. The log transformed variables were plotted again, and t h i s time linear relationships were obtained. 14 4.2.2 Transformations Of V a r i a b l e s In order to explore the best v a r i a b l e s f o r the r e g r e s s i o n a t o t a l of 32 v a r i a b l e s were generated. These were the f o l l o w i n g : lcgDEH, lcgHT, lcgBTBH , HT*DBH, HT*BTBH, BTBH*DBH, DBH 2, HT 2, ETEH 2 , EBH 2HT, EBH 2ETBH ,HT2DBH, HT 2BTBH, BTBH2HT, BTBH2DBH, lcgH1*BTEH, logHT*DBH, logBTBH*DBH, logHT 2, logDBH 2, logETBH 2, lcgD EH 2HT, logEBH 2BTEH, logHT 2 BTB H, logHT 2DBH, logBTBH 2DBH, and lcgBTEH 2HT. The v a r i a b l e s were put through Dr. A. Kozak's stepwise e l i m i n a t i o n program (MREG) to s e l e c t the t e s t ones.. The logarithms of the transformed v a r i a b l e s d i d not improve the equation and were u s u a l l y e l i m i n a t e d at the f i r s t few steps. Thus the logarithms of the transformed v a r i a b l e s were not used. Nineteen s p e c i e s groups were used f o r t h i s e l i m i n a t i o n process. The commonly s e l e c t e d v a r i a b l e s were : logBTBH, logHT, logDBH, EBH 2, HT*EBH, EEH*BTEH, HT 2, BTEH 2 and HT*BTBH. 4.2.3 Stepwise E l i m i n a t i o n The procedure of e l i m i n a t i o n was t r i e d again with o n l y the sguares, c r o s s - p r c d u c t s and l o g s of the independent v a r i a b l e s a g a i n s t the dependent v a r i a b l e logV. T h i s time a l l the 32 sp e c i e s groups were used. Ihe f r e q u e n t l y s e l e c t e d v a r i a b l e s were DBH 2, logDBH, HT*DEH, logHT, and logETBH. From these v a r i a b l e s the f o l l o w i n g 4 best eguations (in terms cf low b i a s and standard e r r o r of estimate) were chosen:-15 {4.1} logBVOL = a +b logBTEH + c log DEH + d logHT {4.2} logBVOL =a +b logBTEH +c logDBH + d logHT + e DBH2 {4.3} logEVCL =a +b logETEH +c logDBH + d logHT +e HT.DBH {4.4} logEVOL =a *b logBTEH +c logDBH +d logHT + e HT.DBH + f DBH2 The 4 equations were tested on a l l species groups for bias and standard e r r c r . 4.2.4 Mas And Standard Error for each species group the diameter range was divided into 10 egual sections in order to observe the variations due tc size cf the trees. Within each diameter class the bias i s computed for each tree and then summed. Eias i s defined as the difference between the predicted tark volume and the actual bark volume, A {4.5} Eias, E = Xi - Yi A where Yi and Yi are the predicted and actual value of t o t a l bark volume for the i t h tree. The t o t a l cumulative bias i s computed for each diameter class within each species group. The predicted values are f i r s t tack transformed from t h e i r logarithms to the o r i g i n a l units of cubic meters (m3). The standard error of estimate for each species group i s computed using the following formula: 16 {4.6} SEE = Z (Yi -YI; i=1 n - p - 1 where n i s the number of observations and p i s the number of independent variables in the equation. 4.3 Besuits And Discussion for each species group the 4 best equations 4.1, 4.2, 4.3 and 4.4 were ranked according to their standard error ( f i r s t being the lowest). The eguation gaining the highest number of f i r s t place rankings among the various species groups was considered to be the best. The ranks of the 32 species groups for the 4 equations are given i n Table 4. Equation {4.4} has 14 f i r s t ranks, 9 second ranks, 6 t h i r d ranks and 4 fourth ranks among the 32 species groups.. Thus i t can be seen that Equation {4.4} i s the best choice.. 17 Table 4. Rank of Standard Errors of Estimate for the 4 Best Bark Volume Eguations i n the 32 species groups* Equ.No. {4.1} 4 + {4.2} I 7 {4.3} {4.4} Rank F i r s t 14 Second 10 Third Fourth 13 18 Table 5. Comparison of Standard Error of Estimate of the 4 best Eguations i n selected species groups.. Species DF1M CE1M I +~ EI12 MBA A Equation Number -+ + +-J L i i J l (in m3) 0.6988 C.2537 0.0081 0.0038 (in m3) 0.7049 0.4802 0.0074 0.0036 (in m3) -+-0.6975 0. 248 1 0.0078 0.0035 (in m3) 0.6913 0.2324 0.0067 0.0042 The standard errors of the 4 best eguations in 4 representative species are given i n Table 5. The standard error of estimate for Eguation {4.4} ranges from 0.6913 m3 with 603 trees for Dcuglas-fir to 0.0042 m3 with 139 trees for maple. The standard error and average bias by diameter class for birch and Douglas-fir using Eguation {4.4} are given i n Tables 6 and 7. The biases and standard error of estimate of the 4 eguations in a l l the species grcups can be found i n APPENDICES 19 II AND I I I . 20 Table 6. Average Eias and Standard Error of Estimate (by DBH class of the best Bark Volume Equation i n Birch (BI12) using Eguation {4.4} . Diam. class No. of trees Bias ( cm) ( m3) 5.0 to 10.0 16 0.0000125 10.0 to 15.0 49 -0.0000979 15.0 to 20.4 104 0.0000355 20.4 to 25.0 69 -0.0000855 25.0 to 30.0 44 0.0000272 30.0 to 35.0 16 0.0002125 35.0 to 40.0 2 -0.023 40.0 to 45.0 1 -0.007 45.0 to 50.0 0 -0.000 50.0 to 55.0 1 -0.018 Total 302 Average 0.0001605 (unweighted) Standard error of estimate = 0.00674 cm Equation {4.4} was selected as the best because cf low standard error of estimate. The average bias per diameter class was examined using the best equation. The range of average bias per species i s from 0.000 1 m3 for birch to 0.036 m3 f o r Douglas-f i r . As shown, the bias i s low for our chosen equation, thus making i t a good estimator for bark volume. 21 Table 7. Average Eias and Standard Error of Estimate(by DBH class for the best Bark Volume Eguation in Douglas Fir(DE1M) using Eguation {4.4}. Diam. class ( cm) No. of trees Bias ( n 3) 10 31 52 73 94 to to to to to 115 to 136 to 157 to 178 to 3 1 52 73 94 115 136 157 178 228 45 136 174 105 67 30 25 14 7 0.0005 -0.0002 0.015 0.086 0.038 -0.24 0.202 0. 126 -0.092 Total 603 Average 0.036(unweighted) Standard error of estimate = 0.69133 cm In general, average bias increases with size of the tree. Thick-barked species, such as Douglas-fir, have the lar g e s t bias while thin-barked hardwoods, such as birch, have the smallest bias. Thus the two species i l l u s t r a t e the range of v a r i a t i o n i n bark volume estimation. The combination of low bias and standard error of estimate makes the chosen eguation adeguate f o r a l l the species studied. The regression c o e f f i c i e n t s f o r Eguation {4.4} for a l l the species groups are given in APPENDIX IV. 22 4.4 Comparison Of The Best Equation Method And The Volume Table Method A simple method for determining bark volume i s tc use volume tables and to assume that the bark volume i s the difference in volume of two boles of the same height. . The difference cf th e i r diameters can be taken as the bark thickness. The metric volume table for B.C. (1976) can be used for this method. This method was compared to the best eguation on 4 species groups , namely Dcuglas-f i r (DF1M) , cedar (CE1M), birch (BI12) and maple (MEAA). The inside 'bark volume table eguations were: lcgV=-4. 348750+1. 692440 logD+1. 181 970 logH (Douglas-fir) lcgV=-4. 536696+1. 907850 logD+1. 1201 60 logH (Maple) logV=-4. 103107+ 1. 743240 logD+0.981729 logH (Cedar) logV=-4. 443142+1.909560 logD+1.052050 logH (Birch) where D i s the diameter outside bark at breast height and H i s the t o t a l height of the tree and V i s the inside bark volume of the tree. Results showed that in a l l cases the volume equations developed in t h i s study were superior and gave a much smaller bias than the volume table method. In par t i c u l a r , for thick-harked species such as Douglas-fir and cedar, the bias i s much larger using the volume table method. The average bias for Douglas f i r for the volume table method was -0.458 m3 compared to -0.036 m3 using the bark volume equation. We observed under-estimation using the volume table method for thick-tarked 23 s p e c i e s . For cedar the va lues of average b i a s were -0 .007 m3 f o r the eguat ion and -0.3150 m 3 f o r the volume t a b l e e s t i m a t e . . 24 Table 8. Comparison of Bark Volume Equation (Eguation {4.4}) and the method of the Volume Table E s t i m a t i o n . . Species DF 1M CE1M MEAA BI12 Average Bias + I Equ.{4.4} Vol^Tab. (in m3) J (in m3) + -0.0367 -4 -0.0077 -0.0009 -0.0003 -0. 4583 -0.3149 + -0.0068 -0.0088 Standard E r r o r (i n m3) 0.6913 0.2324 0.0042 0.0067 Vol^Tabj (in m3) 1.2010 0.6401 0.0108 0.0125 L -In the case of thin-barked s p e c i e s such as b i r c h the d i f f e r e n c e i n the two methods was not as q r e a t . Both methods are reasonable and the average b i a s i s n e g l i g i b l e . . F o r maple the two methods a l s o give very low b i a s , but the volume eguation {4.4} i s s t i l l s u p e r i o r , as i n a l l cases w i t h i n each s p e c i e s group. Table 8 i l l u s t r a t e s the d i f f e r e n c e i n the 2 methods f o r the 4 chosen s p e c i e s . As shown i n the t a b l e the standard e r r o r i s c o n s i s t e n t l y lower f o r Eguation {4.4} . f o r a l l the s p e c i e s considered. A c t u a l l y the standard e r r o r i s h a l f , o r l e s s , of that using the volume t a b l e method of bark volume e s t i m a t i o n . Thus i n e v a l u a t i n g the d i f f e r e n c e of the two methods i t appears 2 5 that the chosen bark volume eguation (Equation [ 4 . 4 } ) i s superior. 2 6 , 4 . 5 Comparison Of Volume Equation And The Method Of Difference Cf P r o f i l e s A t h i r d method of bark volume estimation i s that of u t i l i z i n g the difference of p r o f i l e s . The inside bark diameter p r o f i l e was f i t t e d by Demaerschalk and Kozak(1977) .. With th i s p r o f i l e the volume of the inside bark bole can be determined by integration or summation. In order to predict the outside bark volume a p r o f i l e of the outside bark diameter has to be f i t t e d . The difference between the outside bark p r o f i l e volume and the inside bark p r o f i l e vclume would thus give the bark volume sought. The outside hark diameter p r o f i l e was f i t t e d for a group of chosen species. For every tree sampled both the inside bark volume and the outside bark volume were computed.. The bark volume was predicted as the difference between these two volumes. This method was tested for average bias by comparing the predicted bark volume with the actual bark volume f o r each tree in the sample. The standard error was also computed i n each of the species grcups tested. Table 9 i l l u s t r a t e s the average bias and standard error using t h i s methods and compared i t to the bark volume e g u a t i o n { 4 . 4 } . 27 Table 9. Comparison of Bark Volume Eguation £4.4J and the Method the Difference of the P r o f i l e s i n four selected species groups. I I Average Bias I Standard Error j | Species | Vol.Equ.| P r o f i l e s | Vol.Egu.| P r o f i l e s | | | (in m3) | (in m3) | (in m3) i (in m3) | I 1- + + + 1 1 DF1M J -0.0367 | +0.0395 J 0.6913 j C.7997 | I + + + H i | CE1M | -0.0077 | +0.0034 | 0.2324 j 0.2753 | j 4 + r + i | MEAA | -0.0009 J -C.0006 J 0.0042 } 0.0464 } J BI12 | -0.0003 | -0.0025 | 0.0067 J 0.0599 J I L I I 1 J The p r o f i l e s method frequently has higher averaqe bias and standard error of estimate than the bark volume eguation method. Fcr thick-barked species such as cedar and Douglas-fir the p r o f i l e method seems to over-estimate giving positive average bias. The standard error i s consistently lower for the bark volume eguation in the 4 species tested. Therefore, the bark volume eguaticn appears to be a superior method for determining bark volume. 2 8 5..BARK TAPER PREDICTION 5.1 Objective The objective i s to develop a bark thickness prediction system to accurately estimate bark thickness at any given point in the main tree bole. This model must have the a t t r i b u t e s of low bias and low standard error. 5 . 2 Method 5 . 2 . 1 I n f l e c t i o n Point To represent a l l the data c o l l e c t i v e l y , r e l a t i v e bark thickness (bt/BTEH) was plotted against r e l a t i v e height (ht/HT) The symbols used throughout t h i s part of the t h e s i s are explained in APPENDIX I.. Prom the shape of the plots f o r a l l the species i t seemed reasonable to conclude that the taper can be represented by two curves joined together at some i n f l e c t i o n point.{Examples of plots for 2 species are in APPENDIX V and VI} For t h i s reason an approach similar tc Demaerschalk and Kozak's (1977) model for tree p r o f i l e was adopted. From the plots i t was discovered that i n f l e c t i o n point i s generally at about 20% to 30% of the t o t a l height. 25% was taken as an average figure for use i n the model. A schematic representation of the bark taper model i s i n Figure 1. 2 9 R E L A T I V E B A R K T H I C K N E S S ( bt / BTBH) Ii.9U£i= .!•. Schematic representation of the Bark Taper Model. 3 0 Eark thickness at i n f l e c t i o n point(BTIN) was estimated (since i t was not measured) by quadratic interpolation from the nearest 3 points. This involved solvinq the following eguation: (X -X1) (X -X2) (X -X0) (X -X2) (X -X0) (X -X1) {5.1} Y = Y0 + Y1 + Y2 (X0-X1) (X0-X2) (X1-X0) (X1-X2) (X2-X0) (X2-X1) where (XrY) i s the co-ordinate of the i n f l e c t i o n point and (X0,Y0), (X1,Y1) and (X2,Y2) are the co-ordinates of the nearest 3 points i n the p r o f i l e . Since, by our model, i n f l e c t i o n point i s defined at 25% of the t o t a l height thus X=0.25HT.. It i s necessary to relate bark thickness at i n f l e c t i o n point (BTIN) to some convenient e a s i l y measurable variable. . Bark thickness at breast height (BTBH) was chosen because i t does not depend on the height of the tree and i s available in the data. This has s i g n i f i c a n t p r a c t i c a l application since BTIN cannot be measured d i r e c t l y . The following second degree polynomial was used to f i t the rel a t i o n between tark thickness at breast height (BTBH) and bark thickness at i n f l e c t i o n pcint(BTIN). {5.2} ETIN = b1 + b2 BTBH + b3 BTBH2 where b1,b2 and b3 are constants.. 31 5.2.2 Dp per Taper Model ft non-linear function was used to describe the upper pcrticn cf the tree from i n f l e c t i o n point to the top. . The equation has the form: where bt i s the double bark thickness at any point of the tree above the i n f l e c t i o n point and ht i s the heiqht at that point. HT i s the t o t a l height of the tree, EH i s the distance of the i n f l e c t i o n point from the top r e l a t i v e to HT. .i.e. EH= 0.75 since i n f l e c t i o n point was chosen to be at 25% of the tree height from the ground as shown by the majority of the species studied, and BTIN i s the bark thickness at i n f l e c t i o n point. Constants c1 and c2 are c o e f f i c i e n t s of the model. 32 5.2.3 Lower Taper Model The lower taper model i s used to estimate bark thickness at any pcint below the i n f l e c t i o n point. The model has the form: {5.4} {5.5} . BTIN Coefficient c3 in the Equation {5.4} i s conditioned to make the mcdel predict BTBH without error. The second c o e f f i c i e n t c4 i s conditioned tc make the top and bottom equation continous at the i n f l e c t i o n pcint. 33 5.2.4 Characteristics Of The Bark Taper Model The bark taper model has the following desirable features: 1. The tark thickness at the tree top i s always zero.. 2. The top and bottom models always meet each other at the i n f l e c t i o n point and are continuous at that point. 3. The model predicts bark thickness at breast height (BTBH) without error. 5.2.5 Determination Of Model Coefficients CJj_ C2, C3 And C4 Ihe top eguation can be transformed to logarithmic form as such: (f bt \ /ht/HT\ / ht/HTl j= c1 l n l ]+ ln c2 I 1 BTIN/ \ RH / \ RH / Regressions were f i t t e d for each species f o r the eguation and the c o e f f i c i e n t s c1 and c2 are determined for each species group. With the c o e f f i c i e n t s determined for the upper equation we can then obtain the c o e f f i c i e n t s for the lower model by i t e r a t i o n , so f u l f i l l i n g the requirements of continuity. The Newtcn-Raphson i t e r a t i o n procedure was used to deterrxine the c c e f f i c i n t c4 in the bottom equation i n such a way 3 4 that the curves are continuous at i n f l e c t i o n point. The i t e r a t i v e eguaticn i s : F {5.7} c4 = c4 -DF where F i s a function of the difference of the slope of upper and lower curves and DF i s the f i r s t derivative of the function. The i t e r a t i o n goes on u n t i l F i s minimized to a given tolerance ( 0 . 0 1 was used in this study because i t was found to be s u f f i c i e n t ) . C o e f f i c i e n t c3 i s calculated from c4 as i n eguation {5.5} 5.2.6 Bias And Standard Error The standard error and bias for the taper model were determined for every tree in each species group at each positional height. Eias as defined by the difference between actual bark thickness and predicted bark thickness.i.e.. A {5.9} Eias, B = Yi - Yi A where Yi and Yi are the predicted and actual value of bark thickness at any position in the tree. Standard error at the kth position i s defined as follows: 35 i=1 Nk {5.10} SEEk = N Nk - 1 where Yi and Yi are the prediced and actual value of bark thickness of the i t h tree at the kth position of the tree respectively; and Nk i s the number of data points taken at that p a r t i c u l a r p o s i t i o n 5.3 Besults And Discussion The regression c o e f f i c i e n t s for predicting bark thickness at i n f l e c t i o n point(BTIN) from bark thickness at breast height (BTBH) for a l l the species groups are given i n Table 10. The c o e f f i c i e n t s of determination (R2) range from 0.10 i n lodgepole pine(PLl2) to 0.92 in cottonwood (CT12) . . The model hinges on the concept of the i n f l e c t i o n pcint. Therefore i t i s c r u c i a l tc predict bark thickness accurately at i n f l e c t i o n point. The quadratic equation i s s u f f i c i e n t for bark thickness at i n f l e c t i o n pcint as most species have E 2 values of more than 36 Table 10. Eegression c o e f f i c i e n t s b1, b2 and b3 for estimating bark thickness at i n f l e c t i o n point (BTIN) . . Species b1 b2 b3 R2 CE1M -0. 0321 1. 2348 -0. 0318 0. 57 CE1I -0. 0125 1. 0578 -0. 1265 0. 55 CE2A 0. 0735 0. 9183 -0. 0060 0. 79 HE1M -0. 2440 1. 2373 -0. 0551 0. 68 HE1I 0. 0580 0. 9770 -0. 0585 0. 61 HE2A 0. C339 0. 8883 -0. 0014 0. 79 E A 1A -0. 2284 1. 1195 -0. 0129 0. 78 EA2A -0. 1827 1. 0745 -0. 0 758 0* 83 E A3 A 0. 0568 0. 9699 -0. 0544 0. 73 SE 1M 0. 6777 0. 5321 0. 0207 0. 53 SP1I 0. 6410 0. 3765 0. 0264 0. 43 ME A A 0. 1106 0. 6511 0. 1105 0* 82 SE3A 0. 1481 0. 9307 -0. 0647 0. 57 EI3A 0. 1775 0. 8073 -0. 0961 0. 44 PW2A 0. 101 4 0. 7780 -0. 0151 0. 82 CYAA -0. 2698 1. 5339 -0. 0836 0. 63 L3AA 0. 3393 0. 6905 -0. 0665 0. 25 CT 3 A 0. 0073 0. 7406 0. 0145 0. 9 1 BI12 0. 0746 0. 8981 -0. 0450 0. 76 BI3A 0. 081 1 1. 0489 -0. 1856 0. 45 D123 0. 0228 0. 9462 0. 0058 0. 85 DF 1M 0. 9829 0. 4870 0. 0009 0. 70 SP2A 0. 1905 0. 7353 -0. 0168 0. 43 E112 0. 4037 0. 4748 -0. 0071 0. 10 PYAA -0. 3005 0. 8616 -0. 0251 0. 48 I12A 0. 7482 0. 4829 -0. 0131 0. 45 A12A -0. 0163 0. 8871 -0. 0606 0. 88 DF1I 0. 1540 0. 6091 -0. 0081 0. 80 DF2A 0. 4209 0. 4887 . -0. 0012 0. 83 PW 1A -0. 2133 1. 0529 • -0. 0334 0. 87 CT12 -0. 3482 0. 8944 -0. 0053 0. 92 A3 AA 0. 3450 0. 6267 -0. 0220 0. 71 The f i t t e d c o e f f i c i e n t s c1 and c2 for the upper bark model are given in Table 11. The co r r e l a t i o n c o e f f i c i e n t i s high for a l l species group and remains at about 0.96 for a l l species. Three species, lodgepole pine(PLl2), balsam (BA2A) and alder (A12A), have negative values for logc2 implying that their c2 c o e f f i c i e n t s are small. 3 7 Table 11o Tcp model c o e f f i c i e n t s c1 and c2. Code d log c2 co r r e l a t i o n CE1M 0.767780 0.374638 0. 96 CE1I 0.656 161 0.557772 0.96 CE2A 0.738987 0.411571 0. 96 HE 1 M 0.888873 0.500539 0. 96 HE1I 0.795206 0.425866 0. 96 HE2A 0.857229 0.397729 0.96 E A1 A 0.478904 0.198843 0. 96 EA2A 0.357251 -0.093361 0.96 E A3 A 0.430997 0.103774 0. 96 SP1M 0.348300 0.044876 0.96 SP1I 0.408221 0.257490 0. 96 ME A A 0.709059 0.305399 0. 96 SP3A 0.348780 0.123885 0. 96 PI3A 0.281175 0.091770 0. 96 EW2A 0.492478 0.259419 0. 96 CYAA 0.738635 0.9C5604 0.96 13 A A 0.602141 0.443072 0. 96 CT3A 0.894420 0.400724 0. 96 EI12 0.846730 0.489517 0. 96 EI3A 0.987917 0.815010 0. 96 E123 0.909571 0.902233 0. 96 DF 1M 0.853304 0.348801 0.96 SP2A 0.417896 0.126684 0. 96 EL12 0.284992 -0.C89990 0. 96 EYAA 0.733801 0.337487 0. 96 I12A 0.686951 0.315372 0.96 A12A 0.818329 0.753907 0. 96 DF1I 0.486961 -0.337320 0. 96 DF2A 0.693575 0.046695 0.96 PW 1 A 0.492020 0.279993 0. 96 CT12 0.893870 0.589028 0. 96 A3AA 0.799801 0.833106 0. 96 Coefficients c3 and c4 were calculated for the bottom bark taper p r o f i l e s for each tree such that the top and the the bottom p r o f i l e s were jointed and were continous at the i n f l e c t i o n point. Since the coefficents c3 and c4 are unigue for each tree they are net presented. However, the average bias and standard error for each species group at each of the 7 posi t i o n a l heights of the tree were computed. The average bias 38 and standard error of estimate for birch and Douglas-fir are presented in Table 12 and Table 13. 39 Table 12. Average bias and standard error of estimate of the bark taper model for species Birch (BI12) Position B i a s j i n cm) SEE J i n cm) C.3 m (1ft) -0. 19 0.41 1.3 m (BH) 0.00 0.00 0.2H* 0.02 0.19 0.4H 0.02 0.20 0.6H 0.00 0.21 0.8H 0.00 0.19 1.0H 0.00 0.00 * 0.2H means at 20% of the t o t a l height of the tree from the ground. The average bias along the bole for birch (B112) i s small, ranging from -0.1S cm at the C.3 m butt l e v e l to 0.C0 cm elsewhere in the p r o f i l e . . Birch represents one end of the spectrum for accuracy of the model. On the other end i s coastal Douglas-fir(DF1M) . 40 Table 13. Average bias and standard error of estimate of the bark model f c r species Douglas-fir (DF1M). Position Bias (in cm) SEEJin cm) C.3 m (1ft) 0.82 3.67 1.3 m(BH) 0.00 0.00 0.2H* 0.44 2.05 0.4H 0.26 1.68 C.6H 0.03 1.42 0.8H -0.14 1.16 1.0H 0.00 0.00 * C.2H means at 20% of the t o t a l height of the tree from the ground. Mature Douglas-fir (DF1M) and larch(L12A) have the the largest bias and standard error for the model.. The average biases and standard error for individual species groups are given in APPENDIX VII and VIII. For a l l species the average bias and standard error i s largest at the butt and decreases up the bole. The model has the good advantage of having bias egual to zero at the tree top and at the breast height l e v e l . 41 5.4 Comparison Of Bark Model Kith The Method Of Difference Of P r o f i l e s As mentioned i n the volume section (on page 26) an cuter -diameter p r o f i l e was f i t t e d allowing bark thickness to be computed as the difference between the predicted outside and inside diameter p r o f i l e s (Figure 2). A DIAMETER BARK PR( ZiSi!.I.§ 2_. The Difference of P r o f i l e s method for estimation of bark thickness. 42 Table 14. Comparison of bark thickness estimation using bark model and the method of difference of pr o f i l e s * (species : Douglas-fir(DF1M) ) I Average Bias I Standard Error Position| B._ Model| P r o f i l e s | (in cm) | (in cm) 1 | 0.82 | 0.80 + (- 4 | 0.19 | 0.45 — 4 | 0. 17 | 0. 43 4 (BH) | 0.00 | -0.01 I 4 + 5 | 1.09 | 1.17 *~ 6 [ 0.44 [ 0.59 7 | 0.31 J 0.47 + + _ 0.00 2.44 | 0.26 | 0.43 | 0.17 J 0.33 f + 8 y 9 I 10 | 0.03 | 0.18 *~11 | -0.07 | 0.06 + +-12 | -0.14 | -0.04 T— 13 - 4 | -0.11 1 - 0 . 0 5 -4 14 (Top) | 0.00 | 0.00 1 Model (in cm) 3.67 3. 25 2.59 2.05 1.79 1.68 1.49 1. 42 1.26 1.16 + _ 0.90 0.00 P r o f i l e s (in cm) 3.93 3.84 3.46 2.93 2.45 2.04 1.82 1.69 1.53 1.39 1.18 1.1 1 0.87 0.00 This method of predicting bark thickness was compared to the bark thickness model. The average bias and SEE of this method were computed f o r 4 chosen species groups and then compared to the bark model. Table 14 i l l u s t r a t e s the average bias and standard error for Douglas f i r and Table 15 f o r birch.. 43 Table 15. Comparison of bark thickness estimation using the bark model and the method of the difference of p r o f i l e s , (species : Birch (BI12) ) 5 I 6 B. Model (in cm) -+ | 0.20 | 0.20 4 (EH) | -0.02 | 0.00 | 0.03 - 4 -| 0.03 | 0.16 I 0.07 J 0.03 I 0. 12 -+-0.00 I -+-0. 13 I 0. 13 I 0. 13 7 | 0.01 | 0.12 | 0.16 8 | 0.01 J 0.11 | 0.17 P r o f i l e s (in cm) I I Averacje Bias 1 Standard Error Position 1 B._ Model I P r o f i l e s ] | (in cm) j (in cm) | *~1 { -0.05 | -0.01 2 J -0.07 J -0.03 1- -+ 0. 24 0.22 0.21 0.19 0.21 0.21 0.19 0. 19 9 I 10 I 11 I 12 ( 0.00 J 0.02 | 0.01 I 0.02 I 0.09 | 0.05 I 0.19 | 0.18 0. 19 0. 17 | 0.02 I -+-0. 18 0. 17 | 0.02 J 0.16 0.16 I -+-! - X -13 | 0.00 14 (Top) | 0.00 | -0.03 -+ | 0.00 0. 14 0.00 As observed in Table 14 the average bias for the difference cf p r o f i l e s method and the bark model i s very s i m i l a r . . In some cases the difference^ method actually has a smaller average bias. However, the standard error of estimate i s consistently lower i n the bark model. Also, the difference method does not have zero bias at the breast height l e v e l for most species, including 4 4 Douglas-fir. This i s because the difference method depends on a regression to predict diameter inside bark from diameter outside bark for the inside bark p r o f i l e . Although t h i s regression i s guite accurate i t can not be perfect. In the case cf birch the average bias i s again very similar for both methods. Although at some point the bias i s lower for the differece method the standard error i s consis t l y lower i n the bark model throughout the entire p r o f i l e of the bole. The bias at breast height i s 0.07 cm for the difference method instead of 0.00 cm as i n the case of the bark model. In general comparing the two methods i n terms of average bias we observed similar magnitude of variation..However for the bark model bias i s always zero at breast height unlike the difference of p r o f i l e s method. Moreover, the standard error of estimate i s consistently smaller in the bark model. . Similar r e s u l t s are found in a l l the species tested. Douglas-fir and birch were presented to show the range of variation. . The most notable advantage of the bark model i s that i t i s much simpler tc use one set of eguations instead of two.. From t h i s comparison we again found the bark taper model to be s l i g h t l y superior. 4 5 6. CONCLUSION In t h i s age cf scarcity cf energy resources we must look into renewable and readily available sources for our long term supplies. Tree bark has a high heating value and i s renewable and p l e n t i f u l throughout the province. The objective of t h i s study was to estimate tree bark volume and taper in B r i t i s h Columbia tree species.. The volume of tree bark was studied by f i t t i n g bark volume equations based on measurements of the tree made at the breast height. Also a model was sought that could estimate bark thickness accurately at any point i n the tree stem. Such a model was required to have the attributes of low bias and low standard error of estimate. The model developed was to predict bark thickness based on measurements of the tree at the breast height l e v e l . The tree parameters considered were diameter at breast height (DBH) , t o t a l height (HT) and bark thickness at breast height (BTBH). These parameters were considered for both the bark volume and bark taper studies. from this study a general bark volume prediction eguation was selected for the 32 species groups. The equation was selected for low bias and low standard error of estimate using the following independent variables: DBH, DBH2, BTBH, HT and HT.DEH as below: for the 32 species groups tested. The selected equation i s as follows: logEVOL = a + b loqDBH + c logBTBH+ d logHT + e DBH2 + f HT*DBH where BVOL i s the bark volume, DBH i s the diameter at breast height, BTBH i s the bark thickness at breast 46 height, and HT i s the t o t a l height of the tree.. Table 6 and Table 7 show the range in variation f o r bias and standard error of estimate in 2 different species.. Similarly, for taper, a model was found consisting of 2 curves joined together at the i n f l e c t i o n point (25% of the height of the tree), which s a t i s f a c t o r i l y explained bark taper..The biases and standard error of estimate of the bark taper model are shown at various points i n the tree for 2 selected species in Table 12 and Table 13. However, the model s t i l l showed some over estimation at the stump area, p a r t i c u l a r l y in thick-barked species. This was probably due to the fact that we did not have a measurement at the ground l e v e l , the lowest data point being 1 foot, as a result butt f l a r e was exaggerated. I t i s recommended that in future studies bark thickness measurements be taken as near to the ground as possible i n order to reduce t h i s problem. Over most of the p r o f i l e the bias was small. Prediction at breast height and tree top were without bias. Ey using bark thickness at breast height (BTBH) as an additional parameter our accuracy in the bark volume eguation and tark taper mcdel was improved. Bias and standard error were s l i g h t l y higher when using the difference of p r o f i l e s method where BTBH was not included as an independent variable. For future bark volume and taper estimates i t would be valuable for the cruiser to include an estimate of bark thickness while measuring DEH. The most accurate and most e f f i c i e n t of the 3 methods tested for determining bark volume was the bark volume regression method. For bark taper the thickness model developed 4 7 gave superior r e s u l t s . . A future study of bark weight i s suggested. .Bark weight i s a more constant variable than bark volume or bark thickness as i t automatically takes account of a i r spaces and fissures sc far ignored in thickness and volume measurements. Bark weight at a certain moisture content would be an inter e s t i n g variable to consider in future studies of bark. 48 LITERATURE CITED B.C.F.S.1976.Metric volume tables.Ministry of Forests,B.C.. Blackman, T. 1978. Bark pel l e t s are high energy f u e l f o r coal, gas applications. .For. Ind. 105 (2): 48-49. Chang,Y.P. and R.L. M i t c h e l l . 1955. Chemical composition of ccmmcn North American pulpwood barks. Tappi 38 (5) : 315-320. Demaerschalk,J.P.and A.Kozak.1975.Functions to convert DOB at 4.5feet to DOB at 1*3 meters for the commercial tree species of E.C.Information Beport,Inventory Division,B.C.F.S. Demaerschalk, J. . 1 . and A. Kozak. 1977. The whole-bole system: a conditioned dual- eguation system for precise prediction of tree p r o f i l e . Can. Jour. For. Bes. 7 (3): 488-497. Dimitrov, E.T.1976..Mathematical models for determining .the bark volume of spruce in r e l a t i o n to certain mensurational c h a r a c t e r i s t i c s . .For. .Abstracts 37:6281 Dobie,J. and D.M. Wright. . 1975. Con version factors for the forest products industry in western Canada. Canadian Forestry Service, Western Forest Products Laboratory, Vancouver. Information Beport VP-X-97(Bevised). Fcod and Agriculture Organisation of the United Nations. . 1974. Yearbook of Forest Products for 1972. Borne..371p.. Hale,J.D. 1955. Thickness and density of bark trends of variation for six pulpwood species. Pulp and Paper Magazine 56 (12) : 113-1 17 Husc h , A, M i l l e r , A. E. and CD. Beers. 1970. Forest mensuration. 2nd. Ed. Ronald Press. Jchnson,F.A.1966. Bark factors for Douglas-fir. U.S.F*S.Pacific Northwest Forest and Bange Expt.Stat.Research Note PNW-34. Letkeman, R. 1977. Bark u t i l i z a t i o n : how are we doing? Can. For. Ind. 104 (6) : 18-23. Mater,J. 1977. . U t i l i z i n g bark and wocd residues to solve technical problems. Forest Products Research Society.Madison , Wisconsin. 53705.. Mesavage,C.1969.Measuring bark thickness . J.For.10:753-754 -Meyer, A. 1946. Bark volume determination i n trees. J. Forestry. 44: 1C67-1070. Millikan,D.E.1955. Determination of bark volumes and f u e l properties .Pulp and Paper Magazine of Canada..p106-108.. 49 Schneider, M. H. 1977. Energy from forest biomass* For. Chron,. 53:215-218. Smith,J.H.G. and A. Kozak. 1967. Thickness and percentage of bark cf commercial trees of B r i t i s h Columbia..Mimeo. faculty of Forestry, U. B. C. 33p. Smith,J.H.G. and A. Kozak. 1971* Thickness, moisture content and sp e c i f i c gravity for inner and outer bark of some p a c i f i c northwest trees. FPJ Technical Note. For. Prod. Journal. 21(2) :38-4C. Smith,J.H.G. and 0. S z i k l a i . 1979..Present and potential energy yields of the major tree species of B r i t i s h Columbia. Paper prepared for the Northwest S c i e n t i f i c Association Meeting,March 29-30,1979. Staytcn,C.L. and M. Hoffman.1970..Estimating sugar maple bark thickness and volume. USDA. North Central For. .Fxp. .Stat. . Ees. Pap. NC-38. Von Althen,F.W.1964.Accuracy of the Swedish bark measuring gauge.For.Chron.40(2):257-8 5 0 APPENDIX I L i s t Of Symbols The following symbols are used throughout the thesis: BVCL -- Total tark volume of the main bole DBH Diameter at breast height BTIN -- double hark thickness at i n f l e c t i o n point BTBH — double bark thickness at breast height RH — - r e l a t i v e height of i n f l e c t i o n point from tree top (0.75 RHI r e l a t i v e height of i n f l e c t i o n point frcm tree base (0. 25 HT t o t a l height of tree b1 regression c o e f f i c i e n t f o r bark thickness at i n f l e c t i o n point b2 regression c o e f f i c i e n t for bark thickness at i n f l e c t i o n pcint b3 regression c o e f f i c i e n t f o r bark thickness at i n f l e c t i o n pcint c1 — : — c o e f f i c i e n t for top bark taper model eguation c2 c o e f f i c i e n t for top bark taper model- eguation c3 c o e f f i c i e n t for bottom bark taper model eguation c4 c o e f f i c i e n t for bottom bark taper model equation bt double bark thickness at a given point i n the bole ht height cf the tree at that given point i n the bole BH -• breast height top top of the tree APPENDIX II Comparison Of Total (accumulated) Bias In The 4 Chosen Eguations (in Cubic Meters) Species Egu. {4. 1} Egu. {4. 2} Egu. {4.3} Egu. {' DF 1M 21. 92 24. 35 21.21 22. 13 DF1I 1. 04 5.86 0.99 0. 85 DF2A 4.46 -4 1. 16 10.29 10.35 CE1M 23.06 -119.37 5. 21 4.56 CE2A 9. 13 4.04 4.00 3. 85 HE 1 M 6.91 18.04 24.01 4. 26 HE1I -0.05 0.49 0. 37 0. 44 HE2A 4. 85 3.46 3. 33 3.32 E A 1 A 3.46 3.09 0.64 3. 30 E A 2 A 4.53 1. 53 2.40 2. 21 EA3A 0. 46 13.34 0. 35 1.27 SP1M 9.77 9.17 -7. 96 9.00 SP1I -0.62 0.27 0.22 0.09 SP2A -1.33 7.01 8.88 8. 24 SE3A 1.85 1.71 2. 42 2.09 EL12 1.32 2. 12 2. 37 2. 22 PI3A -0.08 0. 57 0.49 0. 58 CE1I -0.08 0. 16 0. 07 0. 16 PW1A 0.03 0.30 0. 35 0. 31 PW2A 0. 24 0.76 0. 61 0. 62 CY AA 3. 10 0. 43 0.02 0. 22 L12A -9 . 7 8 4.05 3. 70 4. 27 13 A A 0.01 0.07 0.05 0. 06 CT12 2.93 -0.81 1. 58 0. 50 CT3A 5. 51 3.00 2.77 2. 77 BI12 0. 02 0.09 0. 14 0. 09 EI3A 0.07 0.07 0.07 0. 07 A12A -0.02 0.58 0. 70 0. 70 A3 A A -0. 03 0.76 0.97 0.96 ME A A 0.04 0.02 0.02 0. 12 D123 0.30 0.31 0. 35 0.33 P Y AA 1.24 2.68 2. 36 2. 55 APPENDIX III Comparison Of Standard Error Of Estimate In The 4 Chosen Equations (in Cubic Meters) Species Equ. {4. 1} Equ. {4. 2} Equ. {4.3} Equ.{4. DF 1M C.698 0.704 0.697 0.691 DF1I 0. 047 0.055 0. 047 0. 04 9 DF 2 A 0.087 0.093 0.088 0. 085 CE 1M 0. 253 0. 480 0. 248 0.232 CE2A 0.092 0.087 0. 086 0.086 HE 1M 0. 140 0. 146 0. 151 0. 132 HE1I 0. 019 0.019 0.019 0.018 HE2A 0.077 0.076 0.076 0.076 E A 1 A 0. 103 0. 103 0. 103 0.104 EA2A 0. 023 0.022 0.022 0.023 EA3A 0. 105 0.035 0.015 0.015 SP1M 0.329 0. 332 0.371 0. 30 7 SP1I 0.033 0.031 0.033 0.031 SP2A 0.040 0.039 0.040 0.041 SP3A 0.015 0.015 0.015 0.015 EL 12 0.012 0.012 0.012 0.012 EI3A 0.008 0.008 0. 008 0.008 CE1I 0. 015 0.016 0.015 0.016 PW 1 A 0.06 1 0.059 0.059 C. 060 PW 2 A 0.041 0.038 0.038 0.044 CYAA 0.125 0. 119 0. 1 19 0. 117 L12A 0. 134 0. 105 0. 106 0.105 L3AA 0.006 0.006 0.006 0.006 CT12 0.134 0. 191 0. 158 0. 147 CT3A 0.088 0.079 0.080 0.080 BI12 0.008 0.007 0. 007 0.006 BI3A 0.006 0.006 0.006 0. 006 A12A 0.016 0.016 0.016 0.016 A3 AA 0.020 0.018 - 0.017 0.017 MBAA 0.003 0.003 0.003 0. 004 D123 0.010 0.010 0.010 0.010 EY AA 0.092 0.092 0.092 0.091 53 APPENDIX IV The Regression Coe f f i c i e n t s Of Bark Volume Eguation £4.4] Species a b c a e f DF 1M -4. 25738 0.816335 1.44451 0.478939 -1. 302E-5 4.505E--6 DF1I -4. 14634 C.7C7811 1.36476 0.590412 4.813E-5 -2. 415E--5 DF2A -4. 19463 0.657427 1.46899 0.567035 -2.476E-5 6. 105E-6 HE 1M -4. 01229 0.997982 0. 9801 1 0.659254 2.371E-5 3.851E--6 HE2A -4. 31539 1. 189510 1.03225 0.663753 6.656E-6 2. 150E-6 HE 1 M -4. 31549 0.923662 1.31274 0.653487 -2.139E- 5 9. 498E--6 CE1M -4. 39978 1. 192520 1. 09517 0.739025 -6.100E-5 2.450E -5 CE2A -4. 34103 1.004810 1. 20312 0.817377 3.238E-8 1.949E--6 B A 1 A -4. 37409 0.852746 1.38140 0.691159 1.711E-5 -7.073E -6 EA2A -4. 20773 C.901440 1. 19954 0.719798 6.494E-5 -2.623E -5 EA3A -4. 34006 1.030070 1.20059 0.520085 5.206E-5 -7.506E -6 SP 1M -3. 95780 C.925515 1.02844 0.551857 1.755E-5 -3.780E' -6 SP1I -4. 49286 1.129120 1.21729 0.505752 -3.917E- 5 -4.732E -7 SE2A -4. 29314 0.886590 1.2891 7 0.656907 3.08 9E- 5 -3.076E -5 SP 3 A -4. 10998 0.747257 1.30010 0.576554 1.002E-4 -5.611E -5 EL12 -4. 31356 C. 880647 1. 30134 0.587869 6.474E-5 -7.014E -5 PL3A -4. 64824 0.971719 1.55090 0.524239 -1.999E-4 -1.316E -5 CE1I -4. 47989 1. 215312 1. 14064 0.616029 -6.158E- 5 -2.428E -6 PW 1A -4. 41537 0.873283 1.38473 0.618279 2.849E-5 -1.757E -5 PW2A -4. 21924 1. 1 04490 1.04068 0.837029 -1.095E-4 4.219E -5 CY A A -4. 30319 1. 206450 1.02430 0.732134 -6.094E-5 4.333E -5 L12A -4. 23512 0.80 1232 1.42631 0.588780 -5.265E- 5 - 1.216E -5 L3AA -4. 94725 1.634780 1.24165 0.452423 -8.565E-4 3.369E -4 CT12 -4. 02378 0.527363 1.36546 0.648263 1.907E-4 -5.202E -5 CT3A -4. 04569 0.821667 1.09353 0.931462 3.82 9E-6 8.902E -6 E112 -3. 96798 0.639895 1. 21861 0.817264 4.471E-4 -2.698E -4 BI3A -4. 08220 0.763907 1. 12568 0.523373 6.346E-4 -2.326E' -4 A12A -4. 428.94 0.771743 1. 54479 0.625162 7.754E-6 -5.934E -5 A3 AA -4. 44502 0.663185 1.67994 0.468034 1.404E-5 -3.828E -5 ME A A -4. 60207 1. 149170 1.21712 0.567370 1.145E-4 -4.240E -5 D123 -4. 43023 1. 162410 1. 12593 0.722904 8.074E-5 -4.240E -5 PY AA -4. 10194 0.666718 1. 41570 0.596789 -1.785E-5 4.951E--6 SCATTER PLOT OF RELATIVE BARK THICKNESS AGAINST RELATIVE HEIGHT KEL.BT 2.9400 + 2 2 ( COASTAL MATURE DOUGLAS-FIR{OF 1M) ) 2.6133 2* + 2 2 . 2 8 6 7 + * 1.9600 1.6333 1.3067 .98000 .65333 ,32667 0. 2 +**2 6* 62* + X* 68*2 X5* + *X4* 3X63 2XX + 3X32 2X84 3X832 + 3XX4 XX 2* 5X9* +3XX7* XXX2* 0. 9XX7** * * +XXX42 . 3* * * XXX2 *** * * XXX 2 3 2 *2 2 * * +9XXXXXXX84384352* * ** ** * 5XX *X923 2** 2723 2 2 * * *** * XXX* XX844** 48335** 22 ** *2 * * +2X5 2XXX233* 586232* 8 5* 4* 62 3** 3 *X3 XX8*23 4XX252* 43335* 52224 * 22 2 * *** X6 3XXX73* 2XX3* * 9XX52 6X7* *5* * 3 * + 292 XXX92 6XXX2 2 *XXX4** XX72* XX4 *8 3 ' .'3 *3 XXX73 * 5XXX532 9XX63* *XX732 XX 2* X3 * *5 2 2 * XXX3 3XXX4 XXXX2 XXX4 * XX63 *XX2 5X* 52 + XX64 *2XX66 * XXX8 * XXX42 *XX7* 2XX4 7X5 65 6254 2XXX* * XXXX* *XXX3 XX65 **XX2* XX4 X4 2 ** * 2X77* 5XX5 2 X X X 2 * XX8* *4XX4 *XX8 XX 8 + * * *4 3 8X45* 9X92* 2XXX2 3XX3 *XX6** *XX2 * X2 2 2*4 94** XX 3 **XX2 XX6 2XX 2 X2 * * 5*2 3X7 * **XX5* *XX* 3X7* ** ** X3* 4* XX4 * XX* 3 * 3X7* * XXX 6XX + * 3 • 222 22 .44444 .66667 .88889 .11111 .33333 .55 556 • 77778 X + REL.HT I.0000 r 3 SCATTER P L O T O F R E L A T I V E B A R K T H I C K N E S S V S . 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H T I . 0 0 0 0 5 6 APPENDIX VII The Biases Of The Bark Taper Model For A l l Species Groups (in Centimeters) Sp.Grp.. 0.3m E.H. 0.2H.. 0. 4H 0.6H 0.8H 1.0H CE1M 0.68 0.00 -0.38 -0. 33 -0. 26 -0. 13 0. 00 CE1I 0. 18 0.00 -0.02 -0.01 0.01 0. 03 0* 00 CE2A 0.49 0. 00 0.04 0. 03 0. 05 0. 06 0. 00 HE 1M 0. 18 COO -0.01 0.01 -0.03 -0.07 0. 00 HE1I -0.05 0.00 0.04 0. 06 0.05 0. 02 0. 00 HE2A -0. 10 0.00 0.09 0. 06 0.00 -0. 03 0. 00 E A1 A 0. 10 0.00 -0.01 -0. 02 -0.07 -0. 08 0. 00 EA2A -0.03 COO -0.03 -0. 05 -0.01 0. 01 0. 00 EA3A -0.35 0.00 0.03 0. 02 0.05 0. 06 0. 00 SP1M 0. 42 0. oc -0.02 0. 03 -0.01 -0. 01 0. 00 SP1I 0. 04 0. 00 -0.02 -0. 02 -0.03 -0. 04 0. 00 MB AA .-0.05 c o o 0.03 0. 01 -0.02 0. 02 0. 00 SP3A -0. 32 0.00 -0.01 -0.01 0.00 0. 01 0. 00 PI3A -0.20 0.00 0.00 0.01 0.00 0. 01 0. 00 EW2A -0.36 0.00 0.04 0. 01 -0.01 -0. 02 0. 00 CYAA 0. 22 0.00 -0. 28 -0. 30 -0. 29 -0. 14 0. 00 L3AA -0.34 c o o 0.05 0. 04 0.03 0. 06 0. 00 CT3A -0.01 0.00 0.11 0.04 -0.03 0. 09 0. 00 BI12 -0.19 0.00 0.02 0. 02 0. 00 0. 00 0. 00 BI3A -0. 19 c o o 0.02 0. 00 -0.01 -0. 04 0. 00 D123 -0.08 0.00 0.00 0.01 0. 02 0. 01 0. 00 DF1M 0.82 0. 00 0. 44 0. 26 0.03 -0. 14 0. 00 SP 2 A -0.26 c o o 0.05 0. 03 0. 02 0. 00 0. 00 PI12 -0.35 c o o 0.06 0. 05 0.03 0.01 0. 00 PYAA -0.37 0. 00 0. 43 0. 26 0. 12 0. 01 0. 00 112 A -4.66 0.00 0.32 0. 13 -0.05 -0. 12 0. 00 A12A -0.14 c o o 0.06 0. 01 -0.01 0. 00 0. 00 DF1I -0.29 c o o 0. 18 0.06 -0.01 0. 02 0. 00 DF2A -0.22 0. 00 0.40 0. 13 0.03 0. 00 0. 00 PW1A -0. 14 C. 00 0.04 0. 05 -0. 01 0. 01 0. 00 CT12 0.41 0.00 0.12 0. 05 0.00 0. 10 0. 00 A3 AA -0.77 c o o 0.09 0.03 0. 01 0. 05 0. 00 0.2H means at 20% of tHe t o t a l Height of tHe tree form the ground. 57 APPENDIX VIII Standard Error Of Estimate For The Taper Model (in Centimeters) Sp.Grp. 0.3m B. H. 0.2H 0. 4H 0.6H 0. 8H 1.0H CE 1M 1.31 C.OC 1.23 1.27 1. 12 0. 71 0.00 CE1I 0.49 0.00 0.21 0. 25 0. 28 0. 28 0.00 CE2A 1.27 C.OC 0. 57 0.61 0. 58 0. 51 0. 00 HE 1M 0. 92 0. 00 0.70 0. 85 0. 84 0. 66 0.00 HE1I 0.3 8 0.00 0.27 0. 31 0. 36 0.30 0.00 HE2A 0. 73 0.00 0.48 0. 52 0. 57 0. 53 0.00 E A 1 A 0. 56 0.00 0.51 0.59 0.66 0. 54 0.00 EA2A 0.52 0.00 0.31 0.31 0. 33 0. 29 0.00 E A 3 A 0.59 c o o C. 31 0. 33 0.30 0.28 0.00 SP 1M 1.01 0. 00 0.66 0.67 0.65 0. 46 0.00 SP1I 0. 54 c o o 0. 25 0.27 0.26 0. 27 OiOO ME A A 0.20 0.00 0. 13 0. 17 0. 18 0. 16 0.00 SE3A 0.49 c o o 0.29 0. 28 0.27 0. 26 0.00 PL 3 A 0.34 0. 00 0.21 0. 20 0. 19 0. 18 0.00 PW2A 0.60 c o o 0.24 0. 27 0.28 0. 24 0.00 C Y AA 0. 46 0.00 0.72 0. 78 0.89 0. 56 0.00 L3AA 0.52 c o e 0.24 0.23 0.26 0. 22 0.00 CT 3 A 0.65 0.00 0.58 0.60 0.71 0. 59 0.00 BI12 0.41 0. 00 0. 19 0. 20 0.21 0. 19 0.00 BI3A 0.37 c o o 0. 19 0. 18 0.22 0. 20 0.00 D123 0.21 C.OC 0. 17 0. 20 0.24 0. 23 0.00 DF 1 M 3.67 0.00 2. 05 1.68 1. 42 1.16 0.00 SP2A 0. 71 0. 00 0.34 0. 35 0.36 0. 33 0*00 PL 12 0. 77 c o o 0. 27 0. 26 0.25 0. 23 0.00 E Y AA 2. 24 C.OC 0.93 0. 86 0. 90 0. 80 0.00 L12A 7. 77 0.00 0.95 0.79 0.71 0. 65 0.00 A12A 0. 73 c o o 0.34 0. 34 0.33 0. 27 0.00 DF1I 1.18 0. 00 0.63 0. 56 0.47 0. 34 0.00 DF2A 2. 36 c o o 0.93 0. 74 0.67 0.57 0.00 PW 1A 0. 71 0. 00 0.30 0. 38 0. 44 0. 40 0.00 CT12 1.68 c o o 0.68 0.69 0.79 0.80 0.00 A3 A A 1.25 0.00 0. 37 0. 35 0.34 0. 29 0.00 0.2H means at 20% of the t o t a l height of the tree from the ground. 

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